When I initially set up my site, I saw that many admins had been complaining about drowning in (useless) notifications so I thought it was wise to let these go to system. If I’m not mistaken, that was the basic idea of having the system user in the first place, no? So my question was basically "how to notify admins only of replies to posts by system user (and PMs).
But I might have to reconsider. But I’m not yet convinced that this is “the way to solve these problems”. To start with, if I’d change the site contact username to myself now, would I be notified of replies to posts owned by system?
Another thing is that the messages would still go to one admin whereas it would make sense to have all (or selected) admins notified, similar to flags. I could come up with feature requests now, but I’m assuming that this is a somewhat common problem so that it’s probably been solved somehow.